One of the most frequent assertions of the 9/11 "Truth" movement is that the structural steel columns of the
Twin Towers would have resisted being crushed by the falling upper sections. Richard Gage says that one of the characteristics
of destruction by explosives is that "Destruction proceeds through the path of greatest resistance at nearly free-fall
acceleration."

This article presents part of the basic physics of such phenomena (continued in the next two articles), and shows
that, for each tower, the first and subsequent impacts from the falling upper sections
exceeded design safety margins by at least a factor of thirty. Strong as the towers were, they never had a chance
against large falling sections of upper sections of the towers, moving at 20 miles per hour or greater.

The Twin Towers used the "tube within a tube" architectural design. A dense grouping of thick central columns
formed the core, and provided vertical support. The perimeter walls provided both vertical and horizontal
support. The central core was connected to the perimeter walls by the floors themselves, and by the hat truss
on top of the buildings.

The Core and Perimeter Walls

The Hat Truss

The Towers were Mostly Air Inside

The mainstream scientific consensus - what Truthers dismiss as the "Official Story" or "Official Conspiracy Theory" -
has converged on a well-supported explanation of the towers' collapses. The key points of this explanation are that
the planes started fires when they slammed into the towers, each carrying about 10,000 gallons of jet fuel, and going at
hundreds of miles per hour. Several support columns were cut by the airplane impacts, but the hat truss was able to
redistribute loads to the perimeter walls, and keep the buildings standing for a while. The fires weren't hot enough to melt the
steel support beams and trusses, but did soften the trusses enought to where they sagged, and pulled inward on the
perimeter walls.

The perimeter walls were gradually pulled inward, until they snapped under
the strain. This gradual process is not a feature of any controlled demolition.

Once the perimeter walls snapped, the buildings were no longer capable of supporting the upper sections, and these began
to fall. The mainstream view is that the large masses of these upper sections of the towers (about 14 floors for World
Trade Center 1, and around 30 for WTC 2) became unstoppable by the time they had fallen the height of one floor (12 feet, or about
3.8 meters), initiating an "inevitable" progressive collapse of each building.

More recently,
Gage's website, Archbitects and Engineers for 9/11 Truth, has said of the model presented here that "Thomas, like
others who debate Gage, relies on the 'Piledriver Theory'
to explain how the smaller top section of the towers could destroy everything under them. The theory says that a
structural failure due to fires caused the top section to drop one floor, which set off a chain reaction that completely
demolished the buildings. The dynamic force simply overwhelms the building structure below. Gage notes that his theory
ignores the structural resistance supplied by about 300 massive columns in the building."

That means the towers acted like an extremely stiff spring because of their support structures. How stiff? So stiff, that the entire weight of the upper
section (14 floors) would compress the supporting floor's 300 columns by just 8 millimeters - or about a third of an inch.
(The mass of the upper part of the North Tower, WTC1, was on the order of 58x106 kg. The deflection of the
"spring" is force divided by spring constant, i.e. 58 million kg, times g, divided by 71x109 nt/m.)

Those are some pretty stiff support structures! Here's how applied physics can use the stiffness of these structures
to find the actual dynamic force imparted when the upper sections of the towers fell onto the lower sections below.

The Spring Force Equation. The force required to compress the spring a distance x is
proportional to the distance, and to the spring's stiffness, K:

The Energy stored in the spring is the integral of Force, F, over a given Distance, x:

The Spring Energy Equation: potential energy (mgh) converted to spring compression,
where h is the height that the dropping mass falls,
m is its mass, and g is the acceleration of gravity (9.8 m/sec2):

Solving for xdynamic2:

The dynamic force is then:

Recapping, the dynamic force is :

We tested this equation using a small spring scale, with a spring constant of about 60 nt/m. We dropped several
different kinds of masses on it, from spherical ball bearings, to cylindrical nickels, to loosely-wrapped bags of
rice. But, the reactions happened too fast to see, so we got the help of the high-speed camera operated by Professor
Richard Sonnenfeld of New Mexico Tech Physics Department, and found out how hard the masses were impacting the scale.

Here's how the experiments turned out.

The rice was loosely wrapped in some flimsy mesh used for rice packets at weddings, to investigate
the impacts of objects which themselves are not stiff or tightly bound. We did this to learn about impacts of
objects which may consist of a loose assortment of materials.

The rice packet has a mass of 4.8 grams, and a weight of about 0.17 oz, when measured
at rest (static).

The important thing is that if we know the mass of a falling object, the height it falls, and
the stiffness of the structure it impacts, we can estimate the dynamic force on that structure. For the known
stiffness (Spring Constant, 60 nt/m), Rice Mass (4.8 gm), and Drop Height (2 to 16 inches),
the theoretical curve is a parabola. The dynamic force increases as the square root of the drop height. To increase
the force by two, the height is increased by four times. Here, the ratio of dynamic to static force is shown.

The high-speed camera was used to record the point of maximum deflection, which
was used to find the dynamic force.The dynamic force increased just as had been predicted. Here it is for a drop
height of two inches:

Here it is for a drop height of four inches:

Here it is for a drop height of nine inches:

Here it is for a drop height of sixteen inches:

Now, this equation provides estimates of dynamic force for objects of known mass, falling from
known heights, and impacting on objects of known stiffness. For the massive North Tower, the mass was
58x106 kg (top 14 floors), the stiffness 71x109 nt/m, and
the drop height was 3.8 meters.

The bottom line, for the top section of the North Tower impacting the 95th floor after the
failure of the 96th floor:

Here's the thing: this calculation indeed takes into account the stiffness of the towers and their horizontal floor supports.
Yet, we see that the force of the upper section of tower impacting on the bottom section was about 31 times the static force
- the normal weight of the upper section, at rest.

Buildings are designed to be stronger than needed, so as to be able to endure wind, earthquakes, and so on. Usually,
having the building able to withstand loads a factor of three to five times the actual (static) load is considered
adequate for safety. The first floor is simply overwhelmed with a force of 31 times the static load. But, what are
the next floors hit with? That requires consideration of velocity.

For floors after the first one that was impacted, the falling mass has more energy than
that which was gained falling through one floor height. Now, velocity should be used. Here, kinetic energy is
equated with spring compression energy.

Solving for xdynamic2:

The dynamic force is then:

Recapping, the dynamic force is :

After the failure of the 95th floor, the 94th floor is hit by a section of tower
having 62 million kg mass, and a velocity of 11.8 meters per second (as found here).

The next article in this series derives velocities of the collapses, using conservation
of momentum. When these velocities are used to find the dynamic forces on each floor in turn, the accelerations
are found to vary from 31 g's (1st impact), 41 g's (2nd impact), 47 g's (3rd impact), and increasing up to around
67 g's.

The fact is, when that first floor failed, and the upper section fell on the lower part of each tower,
the huge forces, always greater than 30 times the static loads, would have overwhelmed every floor in turn,
leading to progressive collapse. That's why the collapse was "inevitable" - and that's how a building can,
indeed, crush itself.

It's important to realize that this process does not necessarily "break" the
horizontal floor supports. We know that the perimeter walls peeled off to the sides, and also that tottering remnants of the
core columns - the "Spires" - did not fall until several seconds after the main collapse. What was breaking at
each impact were the connections of the floors to the horizontal floor supports. The floors themselves were quickly
pulverized by the tremendous energy of the collapses.

The "Spires", the tottering remnants
of the central core columns.

Another view of the "Spires".

Richard Gage says the Twin Towers showed "Improbable symmetry of debris distribution," and describes this
as one of the "characteristics of destruction by explosives." He is wrong. The firm Weidlinger Associates Inc. (WAI)
was hired by the ownwers of the WTC to determine if the tower collapses were one or two insurable "events."
If Tower 1 was toppled over by Tower 2, for example, the insurance company would pay for only one event. If the
collapses were independent, however, two events would have to be covered. WAI, in their detailed analysis
titled "World Trade Center - Structural
Engineering Investigation," produced detailed maps of the debris distribution. Because the perimeter
walls peeled off from the walls along the four sides of each building, there was a natural 4-leaf-clover pattern
to the debris pileups. But, there was significant asymmetry. Where a tower leaned over a bit during the collapse,
debris was pushed farther from the towers. Also, the debris footprint of Tower 1 is 20 to 25% larger than that for
tower two, because one plane hit Tower 1 near the 96th floor, while Tower 2 was hit around the 80th floor. Tower 1's
debris had farther to fall, and logically was spread out wider. Incidentally, the reason Tower 2 collapsed first
is that the greater mass of the upper section above the plane impact floors put more load on the stressed structure,
making it fail quicker.

Note the lack of debris distribution symmetry, for both South (above) and North (below) Towers.

The main point of this article is that as the upper sections, composed of a given number of floors,
impacted the lower portions of the buildings, floors were broken quickly by the 30-to-70-times-static-weight of the
impacts, becoming part of the growing mass of the upper section. This fact will be used in the next article
of this series, which looks at each impact as the inelastic* collision of an upper section of N floors
with just one floor of the lower section. This approach is reasonable because each floor was broken one at a time,
joining with the growing mass as it became a juggernaut of destruction.

*In an inelastic collision, the two colliding objects stick together.

Most people can hold a 50-pound bag of rocks, However, the impact of that same bag, dropped from 12 feet up,
would almost certainly be lethal.